1.2 Graphs of Functions

In section 1.2 the class discussed even and odd functions.  Functions can be even, odd, or neither and there is multiple ways to determine this.  A function can be tested for evenness and oddness algebraically or graphically.

A function f is even if, for each x in the domain of f, f (-x) = f (x).

A function f is odd if, for each x in the domain of f, f (-x) = - f (x).

A function is neither even or odd when neither of the above is true.

Reminder: It is very important that you understand and remember the definitions of both even and odd functions.

Graphical Solutions

Even Functions

If the graph of a function is symmetric with respect to the y-axis, the function is even.

Examples:

 Odd Functions

If the graph of a function is symmetric with respect to the origin, the function is odd.

Examples: 

Algebraic Solutions Examples

Is this function even, odd or neither? 

In this case f (-x) = f (x) so the function is even and no more work is necessary.

Is this function even, odd or neither? 

f (-x) doesn't equal f (x) so we check to see if f (-x) = -f (x).

This function is odd because f (-x) = -f (x).

Sources:

Sources for graphs: http://www.dummies.com/how-to/content/how-to-interpret-function-graphs.html

http://sophia.hccs.edu/~susan.fife/1314/GraphsofBasicFunctions.htm

Other information found in textbook.  (Precalculus with Limits A Graphing Approach).
Helpful website: purplemath

1.3 Transformations of Graphs of Functions

In unit 1.3, we discussed transforming the graphs of functions.  Graphs can be transformed by being shifted, stretched/compressed, and reflected along or about the axes.  Different types of transformations can be identified by the equation of the graph depending on the value and placement of c.

SHIFTS
Shifts in the graph of a function, or sliding, occur when c is added or subtracted from the parent function.  Horizontal shifts occur when the value of c is subtracted from the value of x.  Vertical shifts occur when the value of c is added to the y-value of the function.

Vertical Shift Upwards:  with 

Vertical Shift Downwards:   with 

Horizontal Shift Right:  with 

Horizontal Shift Left:  with 

Examples:
                                                            

STRETCHES AND COMPRESSION
Stretching and compressing of functions occurs when the value of c is multiplied by f(x) or the value of x.  Horizontal stretching is the stretching of the graph of the function away from the y-axis, and horizontal compression is the compression of the graph towards the y-axis.  Vertical stretching stretches the graph away from the x-axis, and vertical compression compresses the graph towards the x-axis.
Vertical Stretch:  with 

Vertical Compress:  with 

Horizontal Stretch:  with 

Horizontal Compress:  with 

Examples:







REFLECTIONS
Reflections of function graphs result in mirror images of the original graph, reflected over either the x- or the y-axis.  Reflections over the x-axis are vertical reflections and are achieved by multiplying the y-values of the function by -1.  Reflections over the y-axis are horizontal reflections and are accomplished by multiplying the x values by -1.
Vertical Reflection: 


Horizontal Reflection: 

Examples:







THINGS TO REMEMBER
- Things that happen outside of the parentheses affect the y-coordinates of the graph, and cause vertical changes.
- Things that happen inside of the parentheses affect the x-coordinates of the graph, and cause horizontal changes.
- y = f(x) <--> (x,y) is on the graph of f.
(Source: Mr. Wilhelm's Transforming Graphs of Functions handout)

Source for graphs (http://fooplot.com/)
Source for reference information (http://www.regentsprep.org/Regents/math/algtrig/ATP9/funclesson1.htm)

Additional Information
Additonal information can be found here

1.1 Evaluating a Difference Quotient

In 1.1, we also discussed the difference quotient, one of the basic concepts in calculus.
The difference quotient uses the following ratio:

In order to evaluate the difference quotient for a given function, plug the ratio into the function as presented in the examples below. Remember that h can never equal zero.

Example One:

 

 





Example Two:

 






Click here for additional instruction on using the difference quotient.